King of the Hill's cultural position was always a bit of a strange hybrid. On the one hand, it was created as a companion piece to The Simpsons, used a ton of Harvard Lampoon people on the writing staff, and, especially in the first couple of seasons, had a lot of the dry, multileveled humor we associate with the Lampoon and The Simpsons, the kind of jokes that play on the audience's prior expectations or prior knowledge. An example of this kind of joke is the line in the pilot episode where Hank Hill and his friends discuss Seinfeld; the joke is based on the fact that we, the audience, wouldn't expect these guys to talk about Seinfeld while standing near a pickup truck and drinking beer.
On the other hand, the show was rightly celebrated for being an authentic portrait of Texas, co-created by Texas resident Mike Judge and with many native Texans on the writing staff, it was supposed to be an inside look at a region, and a culture, that wasn't usually accurately portrayed on TV. An early episode of the show summed up this side of the show: Hank entertains a client from Boston (voiced by Billy West), who is disappointed that Hank and the town of Arlen don't live up to his preferred stereoypes.
So King of the Hill, especially early on, was kind of a two-tiered show, with two levels of interpretation for two different segments of the audience. For people who identified with the region and culture it portrayed, it was a sympathetic, insider's look at everyday problems within that culture. For people who were not part of that culture, it was a satire, with the characters never quite understanding why they sometimes came off as looking absurd. It was to Texas what Seinfeld was to New York: simultaneously on the side of its characters and keeping an ironic distance from them. Or as one of the writers summed it up: "For most of the country, it's a really cool, smart show about people they know. For New York and L.A., it's like an anthropological study."
The ultimate expression of this two-level strategy was the first episode of KotH's second season, when it was a bona fide phenomenon (a lot of people forget that King of the Hill was huge for its first two seasons, a much bigger cultural phenomenon than Family Guy ever was). This was called "How to Fire a Rifle Without Really Trying", about Bobby Hill, a surprisingly great shot, entering a father-son fun shoot with Hank, who can't shoot at all due to a mental block caused by the way his father, Cotton, treated him. As co-creator Greg Daniels explains on the DVD commentary track for this episode, it was deliberately conceived as the opposite of the typical sitcom episode about guns; most sitcoms did a gun episode at some point, and nearly all of them revolved around the question of whether the family should own a gun at all, usually climaxing in a scene where a family member almost gets accidentally shot, and the family gets rid of the gun. Daniels decided that because "guns were a fact of life in Texas -- it wasn't a question of whether you had them or not," KotH's approach to the subject would be to treat guns and gun ownership as simply a normal part of life. So Hank owns a gun, buys a gun for his twelve year-old son, fathers enter shooting contests with their kids, and none of this is considered unusual.
But having taken that approach, the episode adds a heavy dose of irony: almost every line in the episode is a satirical jab at the characters for being so comfortable with guns. Nobody expresses a genuinely anti-gun viewpoint in the episode (Hank does mention some statistics about the dangers of guns, but he doesn't really believe them), but unbeknownst to themselves, the things the characters say are meant to point out the absurdity of being so completely used to carrying weapons:
HANK: He must've killed a thousand ducks.
PEGGY: A thousand ducks! Well, that is wonderful! Did you kill any bunnies?
INSTRUCTOR: "I didn't know it was loaded" is not an acceptable excuse. "I wasn't there" or "I never met those people" are better excuses. When I was your age, I used to get so excited about hitting the target that I'd run right out onto the range. That's how I lost this thumb, and later, this eye. If it weren't for the NRA safety guidelines which I eventually accepted, I'd be a stub standing here before you.
DALE: Guns don't kill people, the government does.
BILL: Hank, guns have been around for years. If they were dangerous, I think someone would have said something.
In the early seasons especially, Hank used to be the butt of a lot of jokes about his uptightness and old-fashioned views, and a lot of the jokes were based on the idea that something Hank does seems normal to him but is supposed to seem ridiculous to us, most notably his "weird propane fixation." Yet while Hank often had to moderate or change his views by the end of the episode, his basic values were usually proven right -- more right, certainly, than the parade of arrogant "twig-boys" (usually voiced by David Herman) who interfere with Hank's ability to live his life and raise his son the way he prefers. A famous episode in this style was "Husky Bobby", where Bobby becomes a model for plus-sized children's clothes: Hank is mercilessly made fun of for his discomfort with almost everything about late '90s values and pop culture, but in the end, he turns out to be right to pull Bobby out of modeling. Bobby and other characters argue throughout the episode that it's okay for Bobby to look stupid if it makes him feel good about himself (Peggy: "Being different is the most wonderful thing in the whole wide world"), but when he sees the other chubby boy models get picked on and beaten up, Bobby realizes that his father was right: there is a downside to doing whatever feels good in the short term.
King of the Hill has abandoned some of its ironic detatchment as time has gone on, moving closer to Hank's point of view. The episodes today are less likely to end with Hank having to admit he was wrong; instead other characters usually come over to his side. But there's still a good balance between celebrating and mocking Hank and his views, and that's part of what makes the show work for so many different people: if you're culturally conservative, you can enjoy the show's slams on cultural liberalism; if you're culturally liberal, you can enjoy the show's satire of culturally conservative people.
Another reason for KotH's broad appeal is that Hank himself was carefully constructed to be as appealing as possible across the political spectrum. He is an old-fashioned, culturally conservative guy, but he's not unwilling to change his views when new facts are presented to him. He would not be the sort of hack you read on bad conservablogs or hear on morning talk radio, clinging to his position no matter what the facts on the ground happen to be. So almost anyone can respect a guy like Hank.
Also, Hank is often contrasted with other, less reasonable characters, in order to make it clear that he's basically a reasonable guy. In one episode, his conspiracy-mongering friend Dale suddenly decides that the government is right about everything (when he discovers that the Warren Commission Report might have been right, he abandons all his anti-government beliefs); Dale's usual conspiracy theorizing was meant to show that Hank was, by comparison, what we would now call "reality-based," and compared with Dale's newfound jingoism, we see that Hank is the kind of patriot who doesn't mistake patriotism for unquestioning support of the government. Hank is also fairly middle-of-the-road on the big cultural issues; he attends a mainstream Methodist church with a liberal female minister, and in one memorable episode he fights against a fundamentalist who is trying to get Halloween banned as "Satanic." The only regular character on the show who might fit in with the "religious right" (tm) is Hank's niece Luanne, who conducts Bible-study groups in a bikini and expresses an intention to vote for Bush so she can pray in junior college -- but even her religious affiliations are never really specified, and she eventually joins Hank's fight against the anti-Halloween people.
Hank's wife Peggy claims she's not a feminist, and is uncomfortable with ideological feminists like a character voiced by Ani DiFranco (who, Hank's friend Bill Dauterive memorably notes, is "dressed kinda pro-choice"), but she is functionally feminist on many issues, and strongly proclaims her belief in her status as an equal partner in her marriage:
I am not a feminist, Hank. I am Peggy Hill, a citizen of the Republic of Texas. I work hard, I sweat hard and I love hard and I gotta smell good and look pretty while doing it. So, I comb my hair, I re-apply lipstick thirty times a day, I do your dishes, I wash your clothes and I clean the house. Not because I have to, Hank, but because of a mutual, unspoken agreement that I have never brought up, because I am too much of a lady.
That's sort of the cultural position of King of the Hill: the characters live their lives in ways that we might think of as "liberal," but they have a culturally conservative way of looking at the world. When Hank is out of work, he is proud of Peggy for doing extra work to support the family, but he is angry at Bill for suggesting that Peggy is now the man of the house. It's very true to the way most people live their daily lives: people do what they do; they don't politicize their personal decisions; politics and culture are secondary. And the enemies on King of the Hill are usually the people, liberal or conservative, who try to impose broad political or cultural beliefs upon the way we live everyday life.
Which is why KotH is popular with people of every political stripe: it's essentially an apolitical show, where taking politics too seriously interferes with the ability to do what needs to be done in life. That's why I can't really see it as a guide for Democrats or Republicans or anybody: KotH is about issues that don't have political solutions, and is a sort of reminder that politics is not a major part of most people's day-to-day lives.